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<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML 2.2//EN"><!--Converted with LaTeX2HTML 96.1-h (September 30, 1996) by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds --><HTML><HEAD><TITLE>Poincaré sections</TITLE><META NAME="description" CONTENT="Poincaré sections"><META NAME="keywords" CONTENT="TiseanHTML"><META NAME="resource-type" CONTENT="document"><META NAME="distribution" CONTENT="global"><LINK REL=STYLESHEET HREF="TiseanHTML.css"></HEAD><BODY bgcolor=ffffff LANG="EN" > <A NAME="tex2html182" HREF="node12.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html180" HREF="node5.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html174" HREF="node10.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html183" HREF="node12.html">SVD filters</A><B>Up:</B> <A NAME="tex2html181" HREF="node5.html">Phase space representation</A><B> Previous:</B> <A NAME="tex2html175" HREF="node10.html">Principal components</A><BR> <P><H2><A NAME="SECTION00034000000000000000">Poincaré sections</A></H2><P>Highly sampled data representing the continuous time of a differentialequation are called <EM>flow</EM> data. They are characterized by the fact thaterrors in the direction tangent to the trajectory do neither shrink norincrease exponentially (so called marginally stable direction) and thuspossess one Lyapunov exponent which is zero, since any perturbation in thisdirection can be compensated by a simple shift of the time. Since in many dataanalysis tasks this direction is of low interest, one might wish to eliminateit. The theoretical concept to do so is called the Poincaré section. Afterhaving chosen an (<I>m</I>-1)-dimensional hyperplane in the <I>m</I>-dimensional(embedding) space, one creates a compressed time series of only theintersections of the time continuous trajectory with this hyperplane <I>in a predefined orientation</I>. These data are then vector valued discrete time<I>map like</I> data. One can consider the projection of these(<I>m</I>-1)-dimensional vectors onto the real numbers as another measurementfunction (e.g. by recording the value of <IMG WIDTH=14 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline6689" SRC="img37.gif"> when <IMG WIDTH=15 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline6691" SRC="img38.gif"> passes thePoincaré surface), so that one can create a new scalar time series ifdesirable. The program <a href="../docs_c/poincare.html">poincare</a> constructs a sequence of vectors from ascalar flow-like data set, if one specifies the hyperplane, the orientation,and the embedding parameters. The intersections of the discretely sampledtrajectory with the Poincaré plane are computed by a third orderinterpolation.<P><P><blockquote><A NAME="4630"> </A><IMG WIDTH=237 HEIGHT=478 ALIGN=BOTTOM ALT="figure481" SRC="img40.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figdelayPoincare"> </A> Poincaré surface of section using <a href="../docs_c/extrema.html">extrema</a>: A two-dimensional delay plot of the sequence of maxima (top) and of the time intervals between successive maxima (bottom). without employing the option <TT>-t</TT> <I>time</I>, where <I>time</I> is the number of time steps after the last extremum during which no further extrema are searched for (here: 3), one finds some fake extrema due to noise showing up close to the diagonal of the delay representation. Data: Time series of the output power of a CO<IMG WIDTH=6 HEIGHT=11 ALIGN=MIDDLE ALT="tex2html_wrap_inline6701" SRC="img39.gif"> laser [<A HREF="citation.html#INO">35</A>]. <BR></blockquote><P><P>The placement of the Poincaré surface is of high relevance for the usefulnessof the result. An optimal surface maximizes the number of intersections, i.e.\minimizes the time intervals between them, if at the same time the attractorremains connected. One avoids the trials and errors related to that if onedefines a surface by the zero crossing of the temporal derivative of thesignal, which is synonymous with collecting all maxima or all minima,respectively. This is done by <a href="../docs_c/extrema.html">extrema</a>. However, this method suffers more fromnoise, since for small time derivatives (i.e. close to the extrema) additionalextrema can be produced by perturbations. Another aspect for the choice of thesurface of section is that one should try to maximize the variance of the datainside the section, since their absolute noise level is independent of thesection. One last remark: Time intervals between intersections are phase spaceobservables as well [<A HREF="citation.html#Hegger+">36</A>] and the embedding theorems are thusvalid. For time series with pronounced spikes, one often likes to study thesequence of interspike time intervals, e.g. in cardiology the RR-intervals. Ifthese time intervals are constructed in a way to yield time intervals of aPoincaré map, they are suited to reflect the deterministic structure (ifany). For complications see [<A HREF="citation.html#Hegger+">36</A>].<P>For a periodically driven non-autonomous system the best surface of section isusually given by a fixed phase of the driving term, which is also called a<I>stroboscopic view</I>. Here again the selection of the phase should beguided by the variance of the signal inside the section.<P><HR><A NAME="tex2html182" HREF="node12.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html180" HREF="node5.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html174" HREF="node10.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html183" HREF="node12.html">SVD filters</A><B>Up:</B> <A NAME="tex2html181" HREF="node5.html">Phase space representation</A><B> Previous:</B> <A NAME="tex2html175" HREF="node10.html">Principal components</A><P><ADDRESS><I>Thomas Schreiber <BR>Wed Jan 6 15:38:27 CET 1999</I></ADDRESS></BODY></HTML>⌨️ 快捷键说明
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